Repositório UFAL: Teorema de Decomposição de Cheeger-Gromoll.

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

  ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ ▼❛r❝✐✉s P❡trú❝✐♦ ❞❡ ❆❧♠❡✐❞❛ ❈❛✈❛❧❝❛♥t❡

  ▼❛❝❡✐ó✱ ❇r❛s✐❧ ✶✹ ❞❡ ❉❡③❡♠❜r♦ ✷✵✵✼

  ❆♦s ♠❡✉s ♣❛✐s ❋r❛♥❝✐s❝♦ P❡trú❝✐♦ ❈❛✈❛❧❝❛♥t❡ ❡ ▼❛r✐❛ ❞❛s ◆❡✈❡s ❞❡ ❆❧♠❡✐❞❛ ❈❛✈❛❧❝❛♥t❡✳

  ❖ ❝❛♠✐♥❤♦ ❞♦ ❛♠♦r ♥✉♥❝❛ é s✉❛✈❡✱ ♠❛s ❡❧❡ ♣♦❞❡ s❡r ❝♦♥tí♥✉♦✳✳✳

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❍✐❧ár✐♦ ❆❧❡♥❝❛r✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r ❡ss❡s ❛♥♦s ❞❡ ♦r✐❡♥t❛çã♦ ❛❝❛❞ê♠✐❝❛ ❡ ♦♣♦rt✉♥✐❞❛❞❡s ❞❛❞❛s✳

  ❆♦ ♣r♦❢❡ss♦r ❆❞á♥ ❈♦r❝❤♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ❛♠✐③❛❞❡ ❢♦r♥❡❝✐❞❛s ❞❡ s✉❛ ♣❛rt❡ ❞✉r❛♥t❡ ♦s ❛♥♦s ❞❡ ❣r❛❞✉❛çã♦ ❡ ♠❡str❛❞♦✳

  ❆♦ ♠❡✉ ✐r♠ã♦ ❡ ♣r♦❢❡ss♦r ▼❛r❝♦s P❡trú❝✐♦✱ ♣❡❧♦ ❝✉rs♦ ❞❡ ■♥tr♦❞✉çã♦ à ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❛❞♦ ♥♦ ♣r✐♠❡✐r♦ s❡♠❡str❡ ❞❡ ✷✵✵✼✱ ♣❡❧❛ ♠♦r❛❞✐❛ ♣r♦♣♦r❝✐♦♥❛❞❛ ♥❡st❡s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❡t❝✳

  ❆♦s ♣r♦❢❡ss♦r❡s ❊♥❛❧❞♦ ❱❡r❣❛st❛ ❡ ❋❡r♥❛♥❞♦ ❈♦❞á ▼❛rq✉❡s ♣❡❧❛s ✈❛❧✐♦s❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s ❞❛❞❛s ♣❛r❛ ♠❡❧❤♦r✐❛ ❞❡st❛ ❞✐ss❡rt❛çã♦✳

  ❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆▲✱ ❞❡ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ❛ ❆♥❞ré P✐③③❛✐❛✱ ❆r❧②s♦♥ ❆❧✈❡s ❡ ❊✈❡rs♦♥ ❋❡r♥❛♥❞♦✳

  ❆♦s ♠❡✉s ❛♠✐❣♦s ❆s❦❡r② ❆❧❡①❛♥❞r❡✱ ❈❛✐♦ ❡ ▼❛♥♦❡❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋❆▲ ♣❡❧❛ ❢♦rç❛ ❞❛❞❛✱ ❛ss✐♠ ❝♦♠♦ t❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛ ❉❛r❧✐t♦♥ ❈❡③ár✐♦✱ ●❡❧s✐✈â♥✐♦ ❙♦✉③❛ ❡ ❏♦sé ❊❞✉❛r❞♦✳

  ❆♦ ♠❡✉ ❛♠✐❣♦ ▼ár❝✐♦ ❍❡♥r✐q✉❡✱ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ t♦❞♦ ❡s♣❡❝✐❛❧ ♣❡❧❛s ✐♠♣♦rt❛♥t❡s ❞ú✈✐❞❛s ❡s❝❧❛r❡❝✐❞❛s✱ ❛s q✉❛✐s ❞❡r❛♠ ✜r♠❡③❛ ❛ ✈ár✐♦s ❛r❣✉♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳

  ❙♦✉ ♠✉✐t♦ ❣r❛t♦ ❛♦s ♣r♦❢❡ss♦r❡s ❆❞r✐❛♥♦ ❆❣✉✐❛r ❡ ❊❞✉❛r❞♦ P❡r❞✐❣ã♦ ♣❡❧♦s s❡♠✐♥ár✐♦s ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❡ ■♥tr♦❞✉çã♦ à ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ♠✐♥✐str❛❞♦s✳

  ❆♦ ♣r♦❢❡ss♦r ❆❞❡❧❛í❧s♦♥ P❡✐①♦t♦ ❡ s❡✉s ♦r✐❡♥t❛♥❞♦s ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛♣♦✐♦ ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ✐st♦ ✐♥❝❧✉✐ ❈❧❛r✐ss❛ ❈♦❞á ❡ ❈❧❛✉❞❡♠✐r ❙✐❧✈✐♥♦✳

  ❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆▲ q✉❡ ❝♦❧❛❜♦r❛r❛♠ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛✳

  ❆♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲ ❈◆Pq ✲ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

  ❆ ❉❡✉s ♣♦r t✉❞♦✳✳✳

  

❘❡s✉♠♦

  ❉❡♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ♦ q✉❛❧ ❣❛r❛♥t❡ q✉❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ♥✲ ❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✱ q✉❡ ♣♦ss✉✐ ✉♠❛ ❧✐♥❤❛✱ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ♥✉♠ ♣r♦❞✉t♦ ❘✐❡♠❛♥♥✐❛♥♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭♥✲✶ ✮✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s r❡❛✐s✳ P❛❧❛✈r❛s✲❝❤❛✈❡✿ ■s♦♠❡tr✐❛✱ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✱ ▲❛♣❧❛❝✐❛♥♦✱ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ❋✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥✳

  

❆❜str❛❝t

  ❲❡ ❞❡♠♦♥str❛t❡ t❤❡ ❙♣❧✐tt✐♥❣ ❚❤❡♦r❡♠ ❞✉❡ t♦ ❈❤❡❡❣❡r ❛♥❞ ●r♦♠♦❧❧✱ ✇❤✐❝❤ ❡♥s✉r❡s t❤❛t ❛ ❝♦♠♣❧❡t❡ ❘✐❡♠❛♥♥✐❛♥ ♥✲♠❛♥✐❢♦❧❞ ✇❤✐❝❤ ❤❛s ♥♦♥♥❡❣❛t✐✈❡ ❘✐❝❝✐ ❝✉r✈❛t✉r❡ ❛♥❞ ❛ ❧✐♥❡✱ ❝❛♥ ❜❡ s♣❧✐t ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ t❤❡ ❘✐❡♠❛♥♥✐❛♥ ♣r♦❞✉❝t ♦❢ r❡❛❧ ✇✐t❤ ❛ ✭♥✲✶ ✮✲ ♠❛♥✐❢♦❧❞✳ ❑❡② ✇♦r❞s✿ ■s♦♠❡tr②✱ ❲❡✐t③❡♥❜ö❝❦✬s ❋♦r♠✉❧❛✱ ▲❛♣❧❛❝✐❛♥✱ ❙♣❧✐tt✐♥❣ ❚❤❡♦r❡♠ ♦❢ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✱ ❇✉s❡♠❛♥♥ ❋✉♥❝t✐♦♥s✳

  ❙✉♠ár✐♦

  ■♥tr♦❞✉çã♦ ✾

  ✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ✶✷ ✶✳✶ ❱❛r✐❡❞❛❞❡s ❡ ▼étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❆ ❈♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸ ❆♣❧✐❝❛çã♦ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ❱❛r✐❡❞❛❞❡s ❈♦♠♣❧❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✺ ❈✉r✈❛t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

  ✷ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✷✹

  ✷✳✶ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❆ ❍❡ss✐❛♥❛ ❞❛ ❋✉♥çã♦ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

  ✸ ❖ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ ✸✾ ✸✳✶ ❋✉♥çõ❡s ❞❡ ❇✉s❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❚❡♦r❡♠❛ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✽

  ❮♥❞✐❝❡ ❘❡♠✐ss✐✈♦ ✺✵

  ■♥tr♦❞✉çã♦

  ❖ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡✱ ❛❧é♠ ❞✐ss♦✱ ♣♦ss✉✐ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❡st❡ t❡♦r❡♠❛ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ❈♦❤♥✲❱♦ss❡♥✱ ❡♠

  ✸

  ✶✾✸✻✱ ♣❛r❛ s✉♣❡r❢í❝✐❡s ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ❘ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♣♦r ❚♦♣♦♥♦❣♦✈ ❡♠ ✶✾✻✹✱ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❝♦♠♣❧❡t❛s ♥✲❞✐♠❡♥s✐♦♥❛✐s✱ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❆ ✈❡rsã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐✱ ❞❡✈✐❞❛ ❛ ❈❤❡❡❣❡r ❡ ●r♦♠♦❧❧✱ ❣❡♥❡r❛❧✐③❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✱ ❝✉❥♦ r❡s✉❧t❛❞♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✾✼✶ ♥♦ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr② ❝♦♠ ♦ s❡❣✉✐♥t❡ ❡♥✉♥❝✐❛❞♦✿

  ♥

  ❚❡♦r❡♠❛ ✵✳✶✳ ❙❡❥❛ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦✲♥❡❣❛t✐✈❛✳ ❊♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦

  ❦ ❦

  ❘✐❡♠❛♥♥✐❛♥♦ ❞❛ ❢♦r♠❛ ❘ ✂ ◆✱ ♦♥❞❡ ◆ ♥ã♦ ❝♦♥té♠ ❧✐♥❤❛s ❡ ❘ ♣♦ss✉✐ ❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛✳

  ◆♦ ❡♥t❛♥t♦✱ ❝♦♠ ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡

  ♥

  s❡ ✭▼ ❀ ❣✮ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛✱ ❛ q✉❛❧ ❝♦♥té♠ ✉♠❛ ❧✐♥❤❛ ❡ ❘✐❝✭▼✮ ✕ ✵✱ ❡♥tã♦ ▼ é ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣r♦❞✉t♦ ❘✐❡♠❛♥♥✐❛♥♦ ❞❛ ❢♦r♠❛ ❘ ✂ ◆✳ P♦✐s✱ t❡r❡♠♦s ❘✐❝✭◆✮ ✕ ✵ ❡ ◆ ❝♦♠♣❧❡t❛✳

  ❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛

  ❘✐❡♠❛♥♥✐❛♥❛ ❡ ♣r♦❝✉r❛♠♦s ✜①❛r ❛s ♥♦t❛çõ❡s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s sã♦ ❞❡♠♦♥str❛❞♦s✱ ❛ s❛❜❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❋ór♠✉❧❛ ❞❡ ❲❡✐t③❡♥❜ö❝❦✳

  ✸

  Pr♦♣♦s✐çã♦ ✵✳✶✳ ❙❡❥❛ ❢ ✷ ❈ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ ▼✳ ❊♥tã♦ ✁

  ✶

  ✷ ✷

  ✁ ❥ ❣r❛❞ ❢❥ ❂ ❥ ❍❡ss ❢❥ ✰ ❤❣r❛❞ ❢❀ ❣r❛❞✭✁❢✮✐ ✰ ❘✐❝ ✭❣r❛❞ ❢❀ ❣r❛❞ ❢✮ ✿ ✷

  ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ❍❡ss✐❛♥❛ ❞❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❡ ❝♦♥❝❧✉í♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✿

  ♥

  ❚❡♦r❡♠❛ ✵✳✷✳ ❙❡❥❛ ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛✳ ❙✉♣♦✲ ♥❤❛♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ❞❡ ▼ s❛t✐s❢❛③ ❘✐❝✭▼✮ ✕ ✭♥ ✶✮❦❣✳ ❙❡❥❛ r ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ❛♦ ♣♦♥t♦ ♣ ✷ ▼✳ ❙✉♣♦♥❤❛♠♦s ❛✐♥❞❛ q✉❡ ❛ ❢✉♥çã♦ r é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ ①✳ ❊♥tã♦

  ✁r✭①✮ ✔ ✁r ❦ ✭⑦①✮❀

  ✙ ♣

  ♦♥❞❡ r ❦ ✭⑦①✮ ❂ r✭①✮ ❂ r ✵ ❡ r ✵ ❁ s❡ ❦ ❃ ✵✳

  ❦

  ❋✐♥❛❧♠❡♥t❡✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❞❡♠♦♥str❛♠♦s ♦ ❚❡♦✲ r❡♠❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❈❤❡❡❣❡r✲●r♦♠♦❧❧✳ ❆♥t❡s✱ ♣♦ré♠✱ ❛✐♥❞❛ ❞❡♠♦♥s✲ tr❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛♣r❡s❡♥t❛r♠♦s ♦ t❡♦r❡♠❛ ✜♥❛❧ ❝♦♠ ✉♠❛ ❜♦❛ ♦❜❥❡t✐✈✐❞❛❞❡✳ ❉❡♥tr❡ t❛✐s r❡s✉❧t❛❞♦s✱ ♠♦str❛♠♦s q✉❡ ❛ ❢✉♥çã♦

  ✰

  ❞❡ ❇✉s❡♠❛♥♥ ❜ ❛ss♦❝✐❛❞❛ ❛ ✉♠ r❛✐♦ ✌ é s✉♣❡r✲❤❛r♠ô♥✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♠♣❛r❛çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✳

  ❘❡ss❛❧t❛♠♦s q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❢❡✐t❛ ♥❡st❛ ❞✐ss❡rt❛çã♦ é ❜❛s❡❛❞❛ ♥♦ s❡❣✉✐♥t❡ ❛rt✐❣♦✿ ❊s❝❤❡♥❜✉r❣✱ ❏✳✲❍✳✱ ❍❡✐♥t③❡✱ ❊✳✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ❈❤❡❡❣❡r✲

  ●r♦♠♦❧❧ s♣❧✐tt✐♥❣ t❤❡♦r❡♠✱ ❆♥♥✳ ●❧♦❜✳ ❆♥❛❧②s✐s ❛♥❞ ●❡♦♠❡tr②✱ ✈♦❧✳ ✷✱ ✭✶✾✽✹✮✱ ✶✹✶✕✶✺✶✳

  ❖ q✉❛❧ ❛♣r❡s❡♥t❛ ♥♦✈❛s té❝♥✐❝❛s ❞❡ ❝♦♠♦ ❛t❛❝❛r t❡♦r❡♠❛s ❞❡ ❞❡❝♦♠♣♦s✐çã♦✳

  ❱❛❧❡ ♦❜s❡r✈❛r q✉❡ ♦ t❡♦r❡♠❛ ♥ã♦ é ✈á❧✐❞♦ ♣❛r❛ ✈❛r✐❡❞❛❞❡s q✉❡ ♣♦ss✉❡♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥ã♦ ♥❡❣❛t✐✈❛✳ ❈♦♠♦ ❝♦♥tr❛✲❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❛ ✈❛✲

  ✷ ✷

  r✐❡❞❛❞❡ ❘ ✂ ❙ ❝♦♠ ❛ ♠étr✐❝❛ ❞❡ ❙❝❤✇❛r③s❝❤✐❧❞✳

  ❈❛♣ít✉❧♦ ✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛

  ◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ t❛✐s ❝♦♠♦ ✈❛r✐❡❞❛❞❡s ❡ ♠étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s✱ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✱ ❝✉r✈❛t✉r❛s✱ ✈❛r✐❡❞❛❞❡s ❝♦♠♣❧❡t❛s ❡ ♦✉tr♦s ❝♦♥❝❡✐t♦s q✉❡ ♣r❡❝✐✲ s❛r❡♠♦s ♣❛r❛ ♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳

  ✶✳✶ ❱❛r✐❡❞❛❞❡s ❡ ▼étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s ♥

  ❙❛❜❡♠♦s q✉❡ ❞❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ ✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❳ ❡♠ ▼ é ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ ♣ ✷ ▼ ❛ss♦❝✐❛ ✉♠ ✈❡t♦r ❳✭♣✮

  ♣

  ♣❡rt❡❝❡♥t❡ ❛♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❚ ▼✳

  ♥

  ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ① ✿ ❯ ✚ ❘ ✦ ▼✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱ ♣❛r❛ ❝❛❞❛ ♣ ✷ ①✭❯✮✱

  ♥

  ❳ ❅

  ❳✭♣✮ ❂ ☛ ✐ ✭♣✮ ✭♣✮❀ ❅① ✐

  ✐❂✶ ❅ ❅

  ♦♥❞❡ ❢ ✭♣✮❀ ✿ ✿ ✿ ❀ ✭♣✮❣ é ❛ ❜❛s❡ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡♠ ♣ ✷ ▼ ❛ss♦❝✐❛❞❛

  ❅① ❅① ✶ ♥

  à ①✱ ❡ ☛ ✶ ❀ ✿ ✿ ✿ ❀ ☛ ♥ sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ ❯✳ ❈♦♠ ❛❜✉s♦ ❞❡ ♥♦t❛çã♦✱ ❡s❝r❡✈❡♥❞♦ ❢ ❛♦ ✐♥✈és ❞❡ ❢ ✍ ①✱ ♣♦❞❡♠♦s t❛♠❜é♠ ♣❡♥s❛r ❡♠ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❳ ✿ ❉ ✦ ❋ ❞♦ ❝♦♥❥✉♥t♦ ❉ ❞❛s ❢✉♥çõ❡s

  ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ ▼ ♥♦ ❝♦♥❥✉♥t♦ ❋ ❞❛s ❢✉♥çõ❡s ❡♠ ▼✱ ❞❡✜♥✐❞❛ ♣♦r ❳✭❢✮✭♣✮ ❂

  ❅❜

  ❥

  ❜

  ✐

  ❛

  ✐

  ❳

  ❥

  ✰ ❳

  ❅❢ ❅① ❥

  ❅① ✐ ✓

  ❥

  ✐

  ✷

  ✒ ❛

  ✐

  ❳

  ❥

  ❳

  ✓ ❂

  ❥

  ❅①

  ✐

  ❢ ❅①

  ✷

  ❅

  ❢ ❅① ✐ ❅① ❥

  ❥

  ✷

  ❅① ✐ ✿

  ✓ ❅❢

  ❜ ❥ ❅❛ ✐ ❅① ❥

  ❅❜ ❥ ❅① ✐

  ✒ ❛ ✐

  ✐

  ❳

  ❥

  ❳❨ ✭❢✮ ❨ ❳✭❢✮ ❂ ❳

  ✿ ❉❛í✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❙❝❤✇❛rt③✱ ♦❜t❡♠♦s

  ❢ ❅① ❥ ❅① ✐

  ❜ ❥ ❛ ✐ ❅

  ✿ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❡♠♦s q✉❡

  ❥

  ❳

  ✐

  ❳

  ❅① ✐ ✰

  ✓ ❅❢

  ❅❛ ✐ ❅① ❥

  ✒ ❜ ❥

  ❥

  ❳

  ✐

  ❨ ❳✭❢✮ ❂ ❳

  ❅

  ✰ ❜

  

  ♣❡rt❡♥❝❡♥t❡s ❛ ❳ ✭▼✮✳ ❊♥tã♦ ❬❳❀ ❨ ❪ ✿❂ ❳❨ ❨ ❳ ♣❡rt❡♥❝❡ ❛ ❳ ✭▼✮✳

  ❳ ✒

  ❥

  ❳

  ✦ ❂

  ❥

  ❅❢ ❅①

  ❥

  ❜

  ❥

  ✥ ❳

  ❉❡ ❢❛t♦✱ ❳❨ ✭❢✮ ❂ ❳

  ❥ ❅ ❅①

  ❥

  ❜

  ❥

  ❡ ❨ ❂ P

  ✐ ❅ ❅①

  ❛

  ✐

  ❊①❡♠♣❧♦ ✶✳✶✳ ❙❡❥❛♠ ❳ ❂ P

  ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠ ▼ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ❳ ✭▼✮✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞❡ ❞♦✐s ❝❛♠♣♦s✳

  ❅① ✐ ✭♣✮✿

  ☛ ✐ ✭♣✮ ❅❢

  

✐❂✶

  ❳

  ❜

  ❅❢ ❅①

  ❥

  ✓ ❂

  ❅❢ ❅①

  

  ❅①

  

  ✒ ❅❜

  ✐

  ❛

  ✐

  ❳

  ❥

  ❳

  ❥

  ❥

  ❅❢ ❅①

  ❥

  ✒ ❜

  ✐

  ❅ ❅①

  ✐

  ❛

  ✐

  ❳

  ❥

  ❳

  ✓ ❂

  ■ss♦ ♠♦str❛ q✉❡ ❬❳❀ ❨ ❪ é ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳ ❉❡♥♦♠✐♥❛♠♦s ❬❳❀ ❨ ❪ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞♦s ❝❛♠♣♦s ❳ ❡ ❨ ✳

  P❛ss❡♠♦s ❛❣♦r❛ à s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿

  ♥

  ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ ♣ ✷ ▼ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❤✁❀ ✁✐ ❂

  ♣ ♥

  ❣ ♣ ✭✁❀ ✁✮ ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❚ ♣ ▼✱ t❛❧ q✉❡✿ s❡ ① ✿ ❯ ✚ ❘ ✦ ▼ é ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ t♦r♥♦ ❞❡ ♣✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ ✐❀ ❥ ✷ ❢✶❀ ✿ ✿ ✿ ❀ ♥❣✱ ❉ ❊

  ❅ ❅

  ✭q✮❀ ✭q✮ é ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ❯✳

  ❅① ❅① ✐ ❥ q

  ❉ ❊

  ❅ ❅ ✐❥

  ❆s ❢✉♥çõ❡s ❣ ✿❂ ❀ sã♦ ❝❤❛♠❛❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠étr✐❝❛

  ❅① ❅① ✐ ❥ ♥ ♥

  ❘✐❡♠❛♥♥✐❛♥❛ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ① ✿ ❯ ✚ ❘ ✦ ▼ ✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝❤❛♠❛❞❛ ✈❛r✐❡❞❛❞❡

  

  ❘✐❡♠❛♥♥✐❛♥❛✳ ❉❡♥♦t❛r❡♠♦s ♣♦r ✭▼ ❀ ❣✮ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ ♥ ❝♦♠ ✉♠❛ ♠étr✐❝❛ ❣✳

  ♥ ✶ ♥

  ❊①❡♠♣❧♦ ✶✳✷✳ ❆ ❡s❢❡r❛ ✉♥✐tár✐❛ ❙ ✭✶✮ ❂ ❢① ❂ ✭① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ ✮ ✷ ❘ ❀ ❥①❥ ❂ ✶❣ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❈♦♠ ❡❢❡✐t♦✱ ❜❛st❛ ❞❡✜♥✐r♠♦s ❛ ♠étr✐❝❛ ❣ ♣♦r ♥ ✶

  ❣✭✉❀ ✈✮ ♣ ❂ ❤✉❀ ✈✐ ✱ ♦♥❞❡ ✉❀ ✈ ✷ ❚ ♣ ❙ ✳

  ❘ ♥

  ❊①❡♠♣❧♦ ✶✳✸✳ ❖ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ ❍ ✳

  ♥ ♥

  ❈♦♥s✐❞❡r❡ ❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❯ ❂ ❘ ❂ ❢✭① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ ✮ ✷ ❘ ❀ ① ♥ ❃ ✵❣✱

  ✰ ♥

  ❝♦♠ ❛ ❝❛rt❛ ✐❞ ✿ ❯ ✚ ❘ ✦ ❯✳ ❉❛❞♦ ♣ ✷ ❯✱ ♣ ❂ ✭① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ ✮✱ s❡❥❛♠

  ♥ ♣

  ✈❀ ✇ ✷ ❚ ❯ ❂ ❘ ✳ ❉❡✜♥✐♠♦s

  ✶ ❤ ♣ ✭✈❀ ✇✮ ❂ ❤✈❀ ✇✐ ✿

  ❘ ✷

  ①

  ♥ ♥

  ❱ê✲s❡ q✉❡ ✭❯❀ ❤✮ ✿❂ ❍ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❞❡♥♦♠✐♥❛❞❛ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦✳ ❊①❡♠♣❧♦ ✶✳✹✳ ❱❛r✐❡❞❛❞❡s ■♠❡rs❛s✳

  ♥ ♥✰❦

  ❙❡❥❛ ❢ ✿ ▼ ✦ ◆ ✉♠❛ ✐♠❡rsã♦✱ ✐st♦ é✱ ❢ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧

  ♣ ♣ ❢✭♣✮

  ❡ ❞❢ ✿ ❚ ▼ ✦ ❚ ◆ é ✐♥❥❡t✐✈❛ ♣❛r❛ t♦❞♦ ♣ ✷ ▼✳ ❙❡ ◆ t❡♠ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ❢ ✐♥❞✉③ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ▼✱ ❞❛❞❛ ♣♦r

  ♣ ♣ ♣

  ❤✉❀ ✈✐ ❂ ❤❞❢ ✭✉✮❀ ❞❢ ✭✈✮✐ ❀ ✉❀ ✈ ✷ ❚ ▼✿

  ♣ ❢✭♣✮

  ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ ❞❢ ♣ é ✐♥❥❡t✐✈❛✱ ❤✁❀ ✁✐ é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ❆s ❞❡♠❛✐s ❝♦♥❞✐çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡

  ✈❡r✐✜❝❛❞❛s✳ ❆ ♠étr✐❝❛ ❞❡ ▼ é ❡♥tã♦ ❝❤❛♠❛❞❛ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣♦r ❢ ❡ ❢ é ✉♠❛

  ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✳

  ♥ ♥✰❦

  ◗✉❛♥❞♦ ❢ ✿ ▼ ✦ ◆ é ✉♠❛ ✐♠❡rsã♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❢ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ❢✭▼✮ ✚ ◆✱ ♦♥❞❡ ❢✭▼✮ t❡♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ◆✱ ❞✐③✲s❡ q✉❡ ❢ é ✉♠ ♠❡r❣✉❧❤♦✳ ❙❡ ▼ ✚ ◆ ❡ ❛ ✐♥❝❧✉sã♦ ✐ ✿ ▼ ✱✦ ◆ é ✉♠ ♠❡r❣✉❧❤♦✱ ❞✐③✲s❡ q✉❡ ▼ é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ◆✳

  ❯♠❛ ❝❧❛ss❡ ✐♠♣♦rt❛♥t❡ ❞❡ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s é ❞❛❞❛ ♥♦ ❡①❡♠♣❧♦ s❡❣✉✐♥t❡✳ ❊①❡♠♣❧♦ ✶✳✺✳ ▼étr✐❝❛ Pr♦❞✉t♦✳

  ♥ ♥ ✶ ✷ ✶ ✷

  ❙❡❥❛♠ ✭▼ ✶ ❀ ❣ ✮ ❡ ✭▼ ✷ ❀ ❣ ✮ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❡ ❝♦♥s✐❞❡r❡ ♦ ♣r♦❞✉t♦

  ✶ ✷

  ❝❛rt❡s✐❛♥♦ ▼ ❂ ▼ ✂ ▼ ❝♦♠ ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣r♦❞✉t♦✳ ❙❡❥❛♠

  ✶ ✶ ✷ ✶ ✷ ✶ ✷ ✷

  ✙ ✿ ▼ ✂ ▼ ✦ ▼ ❡ ✙ ✿ ▼ ✂ ▼ ✦ ▼ ❛s ♣r♦❥❡çõ❡s ♥❛t✉r❛✐s✳ P♦❞❡♠♦s ✐♥tr♦❞✉③✐r ❡♠ ▼ ✶ ✂ ▼ ✷ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♣♦♥❞♦✱ ♣❛r❛ ❝❛❞❛ ✭♣❀ q✮ ✷ ▼ ✶ ✂ ▼ ✷ ❡ ✉❀ ✈ ✷ ❚ ✭♣❀q✮ ✭▼ ✶ ✂ ▼ ✷ ✮✱

  ❤✉❀ ✈✐ ❂ ❤❞✙ ✶ ✁ ✉❀ ❞✙ ✶ ✁ ✈✐ ✰ ❤❞✙ ✷ ✁ ✉❀ ❞✙ ✷ ✁ ✈✐ ✿

  ✭♣❀q✮ ♣ q ✶ ✶ ✷

  ❈♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❙ ✂ ❙ ❂ ❚ ♦✉✱ ♠❛✐s

  ✶ ✶ ♥

  ❣❡r❛❧♠❡♥t❡✱ ❙ ✂ ✁ ✁ ✁ ✂ ❙ ❂ ❚ ✱ ♦s q✉❛✐s tê♠ ✉♠❛ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛

  ✶ ✷

  ♦❜t✐❞❛ q✉❛♥❞♦ ❡s❝♦❧❤❡♠♦s ♥♦ ❝ír❝✉❧♦ ❙ ✚ ❘ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛

  ✷ ♥

  ✐♥❞✉③✐❞❛ ♣♦r ❘ ❡✱ ❡♠ s❡❣✉✐❞❛✱ t♦♠❛♠♦s ❛ ♠étr✐❝❛ ♣r♦❞✉t♦ ❡♠ ❚ ✳ ❖ t♦r♦

  ♥

  ❚ ❝♦♠ ❡st❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝❤❛♠❛❞♦ t♦r♦ ♣❧❛♥♦ ♦✉ t♦r♦ ✢❛t✳ ❚❡♦r❡♠❛ ✶✳✶✳ ❚♦❞❛ ✈❛r✐❡❞❛❞❡ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❞❡ ❍❛✉s❞♦r✛ ♣♦ss✉✐ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✹✼✳

  

  ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ✭▼ ❀ ❣✮ ❡ ✭▼ ❀ ❣✮ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s✳ ❯♠ ❞✐❢❡♦♠♦r✜s♠♦ ✬ ✿ ▼ ✦ ▼ é ❞✐t♦ ✉♠❛ ✐s♦♠❡tr✐❛ s❡

  ❣ ✭❞✬ ♣ ✭✈✮❀ ❞✬ ♣ ✭✇✮✮ ❂ ❣ ♣ ✭✈❀ ✇✮❀ ✽ ♣ ✷ ▼❀ ✽ ✈❀ ✇ ✷ ❚ ♣ ▼✿

  ✬✭♣✮

  ✶✳✷ ❆ ❈♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛

  ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é ✉♠❛ ❛♣❧✐❝❛çã♦ r ✿ ❳ ✭▼✮ ✂ ❳ ✭▼✮ ✦ ❳ ✭▼✮ t❛❧ q✉❡✱ ✽ ❳❀ ❨❀ ❩ ✷ ❳ ✭▼✮ ❡ ✽ ❢❀ ❣ ✷ ❉✭▼✮✱ t❡♠✲s❡

  ✭✐✮ r ❢❳✰❣❨ ❩ ❂ ❢r ❳ ❩ ✰ ❣r ❳ ❩ ✭✐✐✮ r ❳ ✭❨ ✰ ❩✮ ❂ r ❳ ❨ ✰ r ❳ ❩

  ❳ ❳

  ✭✐✐✐✮ r ❢❨ ❂ ❢r ❨ ✰ ❳✭❢✮❨ ✳

  

  ❊♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✱ ❳ ❂ ✱ ♣♦❞❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ✈❡r✱ ♣❛r❛ ❳ ❂

  ❅①

  P P

  ✐ ✐ ✐ ✐

  ① ❳ ❡ ❨ ❂ ② ❨ ✱ q✉❡

  ✐ ✐

  ✥ ✦ ❳ ❳

  ❦

  r ❳ ❨ ❂ ① ✐ ② ❥ ✰ ❳✭② ❦ ✮ ❳ ❦ ✿

  ✐❥ ❦ ✐❀❥

  P

  

❦ ❦

  ❖s ❝♦❡✜❝✐❡♥t❡s ✱ ❞❡✜♥✐❞♦s ♣♦r r ❳ ❳ ❥ ❂ ❳ ❦ ✱ sã♦ ❝❤❛♠❛❞♦s

  

✐❥ ✐ ✐❥

  sí♠❜♦❧♦s ❞❡ ❈❤r✐st♦✛❡❧ ❞❛ ❝♦♥❡①ã♦✳ ❊♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ♦s sí♠❜♦❧♦s ❞❡

  ❈❤r✐st♦✛❡❧ ♣♦ss✉❡♠ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ ✚ ✛

  ❳ ✶ ❅ ❅ ❅

  ♠ ❦♠ ❥❦ ❦✐ ✐❥

  ✐❥ ❂ ❣ ✰ ❣ ❣ ❣ ❀ ✐ ❥ ❦

  ✷ ❅① ❅① ❅①

  ❦ ❦♠ ❦♠

  ♦♥❞❡ ✭❣ ✮ é ❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ ✭❣ ✮✱ ✈❡r ❬✸❪✱ ♣á❣✐♥❛ ✻✷✳ ❉❛❞❛ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ☛ ✿ ■ ✚ ❘ ✦ ▼✱ ✉♠ ❝❛♠♣♦ ❛♦

  ❧♦♥❣♦ ❞❡ ☛ é ✉♠❛ ❢✉♥çã♦ ❱ ✿ ■ ✦ ❚ ▼ t❛❧ q✉❡ ❱ ✭t✮ ✷ ❚ ☛✭t✮ ▼✱ ♦♥❞❡ ❚ ▼ ❂ ❢✭♣❀ ✈✮❥♣ ✷ ▼❀ ✈ ✷ ❚ ♣ ▼❣ é ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ ▼✳

  ❞

  ❙❡❥❛ t ✷ ■✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ t❡♠♦s ❛ ❜❛s❡ ❢ ❣ ❡♠ ❚ t ■ ❂ ❘✳ P♦r

  ❞t ❞ ✵

  ❞❡✜♥✐çã♦✱ ☛ ✭ ✮ é ♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❝✉r✈❛ ☛ ❡♠ ☛✭t✮✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦✿

  t ❞t

  ✁

  ❞ ✵

  ❞☛ ❂ ☛ ✭t✮✳

  ❞t Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡❥❛♠ ▼ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡ ☛ ✿ ■ ✦ ▼ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ ❝❛♠♣♦ ❱

  ❉❱

  ❛♦ ❧♦♥❣♦ ❞❡ ☛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❝❛♠♣♦ ❞❡♥♦t❛❞♦ ♣♦r t❛❧ q✉❡

  ❞t ❉ ❉❱ ❉❲

  ✭❛✮ ✭❱ ✰ ❲ ✮ ❂ ✰ ❀

  ❞t ❞t ❞t ❞❢ ❉ ❉❱

  ✭❜✮ ✭❢❱ ✮ ❂ ❱ ✰ ❢ ❀

  ❞t ❞t ❞t

  ✭❝✮ ❙❡ ❱ ✭t✮ é ❛ r❡str✐çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❨ ❞❡✜♥✐❞♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡

  ❉❱

  ☛✭■✮✱ ❡♥tã♦ ❂ r ☛ ✭t✮ ❨ ✳

  ❞t

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✸❪✱ ♣á❣✐♥❛ ✺✼✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é s✐♠étr✐❝❛ s❡

  ❳ ❨

  r ❨ r ❳ ❂ ❬❳❀ ❨ ❪❀ ✽ ❳❀ ❨ ✷ ❳ ✭▼✮✿ ❆ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ t❛❧ ♥♦♠❡♥❝❧❛t✉r❛ ✈❡♠ ❞♦ s❡❣✉✐♥t❡ ❢❛t♦✿ ✽ ✐❀ ❥ ✷

  ❢✶❀ ✁ ✁ ✁ ❀ ♥❣✱ r ❳ ❳ ❥ r ❳ ❳ ✐ ❂ ❬❳ ✐ ❀ ❳ ❥ ❪ ❂ ✵❀ ✐ ❥

  ❦ ❦

  ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❂ ✳

  ✐❥ ❥✐

  ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝♦♥❡①ã♦ ❛✜♠ r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ▼ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ s❡

  ❳ ❤❨❀ ❩✐ ❂ ❤r ❳ ❨❀ ❩✐ ✰ ❤❨❀ r ❳ ❩✐ ❀ ✽ ❳❀ ❨❀ ❩ ✷ ❳ ✭▼✮✿ ❖ ♣ró①✐♠♦ t❡♦r❡♠❛ é ❢✉♥❞❛♠❡♥t❛❧ ❡♠ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳

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